Melt the butter in a large wide pot over medium heat. Add the onions and cook, stirring occasionally, until soft but not brown, about 10 minutes. Add the apples and cook, stirring, until soft but not mushy, about 10 minutes more.
Sprinkle the flour and nutmeg over the onions and apples. Cook, stirring, until the mixture is light brown and fragrant, 4 to 5 minutes. Add the apple-flavored brandy and continue to cook, stirring, until the pot is almost dry.
Gradually whisk in the broth until well blended. Bring to a boil, reduce the heat to a simmer, cover and cook, stirring occasionally, for 10 minutes. Add the milk and return to a simmer. Whisk in the cheese a handful at a time until completely melted. Season with salt and pepper.
Serve in soup bowls and garnish with more cheese and the chives if using.
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